rail contact model

Engineering Structures 144 (2017) 120–138

Contents lists available at ScienceDirect

Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Probability analysis of train-track-bridge interactions using a random wheel/rail contact model Zhi-wu Yu, Jian-feng Mao ⇑ a b

School of Civil Engineering, Central South University, Changsha 410075, China National Engineering Laboratory for High-Speed Railway Construction, Changsha 410075, China

a r t i c l e

i n f o

Article history: Received 15 September 2016 Revised 3 April 2017 Accepted 19 April 2017 Available online 6 May 2017 Keywords: Random wheel/rail contact interaction Train-track-bridge system Probability density evolution method Stochastic harmonic functions Number theoretic method

a b s t r a c t Based on the random theory of the probability density evolution method (PDEM), a refined random wheel/rail contact model for train-track-bridge systems is proposed. The trace line method with the law of equal slopes is developed to determine the wheel/rail contact locations, which are described by probability density functions on the wheel/rail surface. The number theoretic method (NTM) and stochastic harmonic functions (SHFs) are employed to generate the representative random track irregularities. Considering the excitation of random track irregularities and the wheel/rail contact geometry, the wheel/rail normal contact force and the tangential creep force are accurately calculated and analyzed using the proposed wheel/rail contact model. A high-speed railway case study is presented to reveal the random dynamic characteristics of a train-track-bridge system and the wheel/rail interaction. The results show that the time-varying probability density evolution functions of the wheel/rail contact points can effectively indicate both the phenomenon of wear on the rail surface and its distribution characteristics. Additionally, some significant conclusions are obtained. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The spatially rolling contact model of wheel/rail interaction is the basis of train-track-bridge dynamic interaction systems [1]. The wheel/rail interaction is a complex problem that has plagued researchers and engineers for decades. Many attempts have been made to find different solutions to this problem [2–6]. As a result of these investigations, the wheel/rail forces at the contact points have been calculated. Kalker [7] made the greatest contribution to wheel/rail contact creep theory, and the simplified wheel/rail contact theory is convenient for engineering applications [8]. Subsequently, many improved wheel/rail contact models were proposed, e.g., those by Pombo [9], Chen and Zhai [10], and Shabana et al. [11]. Typically, the coupled dynamic analyses of moving vehicles on a bridge are conducted using a spring-damper as the primary suspension system between the wheel-sets and the bogies [12–14]. Researchers have made continuous contributions to the analysis of train-bridge dynamics, and a number of sophisticated analytical models involving refined wheel/rail interaction have been ⇑ Corresponding author at: School of Civil Engineering, Central South University & National Engineering Laboratory for High-Speed Railway Construction, NO. 22, Shaoshan Road, Changsha 410075, China. E-mail address: [email protected] (J.-f. Mao). http://dx.doi.org/10.1016/j.engstruct.2017.04.038 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

developed [15–21]; the dynamic simulation results of these sophisticated models provide more details than were available previously. For a given profiled tread of the wheel and rail, predicting the locations of the wheel/rail contact points using the current theory is not easy much less when considering the effect of the random dynamic responses of a train-track-bridge system. Due to the complexity of the wheel/rail interaction model, previous investigators usually employed deterministic analysis to calculate this model. However, the wheel/rail interaction is considered to experience random dynamic motion. Thus, one should comprehensively consider the influences of variations in the wheel/rail geometry, variations in the railway cant angle, random track irregularities, and random dynamic characteristics of subgrade and bridges. The random dynamic theory can better reveal the random dynamic characteristics of train-bridge coupled system than deterministic analysis. With the great improvements in computing performance, random theories for train-bridge coupled systems are becoming increasingly accurate and providing deeper understanding of these systems. The Monte Carlo method (MCM) is usually adopted to obtain the random dynamic simulation of train-bridge systems for a large number of samples [22–25] and for probabilistic safety assessment of railway bridges [26,27]. MCM has also been compared with other methods, such as the Latin hypercube (LH) method [28] and enhanced simulation (ES) method [29]. Rocha

121

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

et al. [30] conducted an efficient comparison of methods for the probabilistic safety assessment of high-speed railway bridges. However, huge computational expenses are necessary to provide adequate analysis accuracy, greatly limits the development of MCM in these applications. The pseudo excitation method (PEM), developed by Lin [31], is a convenient and efficient algorithm for random dynamic analysis, e.g., for buildings, bridges [32,33], and the random dynamic analysis of train-track-bridge systems [34]. However, PEM cannot obtain the mean value of a random dynamic response, which is considered a deficiency for simulating trainbridge interaction. The stochastic perturbation method (SPM) [35] is generally employed for the stochastic dynamic analysis of train-bridge systems with varying material properties and mechanical uncertainties. For nearly two decades, the generalized probability density evolution method (PDEM), a random theory developed by Li and Chen [36–38], has been thoroughly investigated for random structural dynamic analysis. Numerous studies have demonstrated that the instantaneous probability density functions (PDFs) and their evolution of arbitrary physical quantities for linear and nonlinear stochastic systems can be successfully captured using PDEM [36] at a higher efficiency than using MCM. PDEM was first successfully applied to the random dynamic analysis of train-bridge coupled systems by Yu and Mao [39,40], and they validated PDEM as a superior and efficient method for the dynamic analysis of railway train-bridge coupled systems. In this paper, taking a realistic wheel tread profile and 60 kg/m steel rail as an example, a refined wheel/rail random contact model based on PDEM is proposed for the dynamic analysis of 3D traintrack-bridge systems. The wheel/rail contact points, which are described using the time-history probability density evolution functions, effectively exhibit the probabilistic characteristics of wheel/rail interaction. The trace line method (TLM) with the law of equal slope is developed to find the wheel/rail contact points, and the dynamic characteristics of wheel/rail interaction are accurately determined. Random track irregularity samples are generated using the stochastic harmonic function (SHF) [41] and number theoretic method (NTM) [37,42]. Finally, the probability density evolution equation of a train-track-bridge system with random wheel/rail interaction is derived, and the instantaneous PDFs of the response are captured. Discussions on the random dynamic responses of vehicles and bridge are presented, and some significant conclusions are obtained. 2. Modeling train-track-bridge systems with random wheel/rail interaction As shown in Fig. 1, the train-track-bridge system for a highspeed railway is composed of vehicles, steel rails, a slab track,

and the bridge substructure. Each of these subsystems is simulated as an elastic substructure, and the interaction between the wheels and the rails is the basis of the train-track-bridge system. Based on the principle of total potential energy with stationary value in elastic system dynamics [43], the 3D dynamic equation of motion for the train-track-bridge system is established with the following submatrix form:

9 2 9 38 38 € _ C v r ðxÞ 0 > Cv > < Xv > < Xv > = = 6 7 € 6 7 0 5 X þ 4 C v r ðxÞ Cr ðxÞ Crtb 5 X_ r 4 0 Mr r > > > :€ ; :_ > ; 0 Ctbr Ctb 0 0 Mtb Xtb Xtb 9 9 8 2 38 Kv r ðxÞ 0 > Kv < Fv ðH; xÞ > < Xv > = = > 6 7 þ 4 Krv ðxÞ Kr ðxÞ Krtb 5 Xr ¼ Fr ðH; xÞ > > > : : ; ; > 0 0 Ktbr Ktb Xtb 2

Mv

0

0

where the subscripts ‘v ’, ‘r’and ‘tb’ denote the vehicles, rails, and track-bridge system, respectively, and ‘x’ denotes the longitudinal coordinate on the rail. Mv , Mr and Mtb represent the mass matrices of the vehicles, rails, and track-bridge system, respectively. As the system mass matrices do not change with the vehicle location, they are not directly related to ‘x’. Kv , Kr ðxÞ and Ktb represent the stiffness matrices of the vehicles, rails, and track-bridge system, respectively; Kv r ðxÞ ¼ Krv ðxÞT represents the wheel/rail interaction varying with time, and Krtb ¼ KTtbr represents the rail-track-bridge interaction. Xv , Xr and Xtb represent the displacement vectors of the vehicles, rails and track-bridge system, respectively; Fv ðH; xÞ and Fr ðH; xÞ represent the random load vectors acting on the vehicles, rails system, respectively. The damping matrices in Eq. (1) have the same form as the stiffness matrices and can be obtained by simply replacing K with C. The symbol H ¼ ðn1 ; n2 ; . . . ; nnpt Þ represents the random variable space of random track irregularities that are involved in the traintrack-bridge system. ni ði ¼ 1; . . . ; nptÞ denotes the discrete point set of random spatial frequencies and random phases that are used for the track irregularity sample generation. npt denotes the total number of random vectors. The track-bridge system is associated with the slab track, bridge girders and piers.

2.1. Wheel/rail interaction model and random load vectors 2.1.1. Assumptions for the wheel/rail interaction Many random excitations occur in a train-track-bridge system, and random track irregularity is considered to be the main one. To consider the influence of random track irregularity and the resulting structural vibrations, the random wheel/rail contact points on the wheel/rail surface should be described using time-history PDFs.

Random track irregularites zwk

wk

xwk

Vehicles

o wk

C

yw

xr

r

r

Bridges

ð1Þ

r

yr

zr

Track

Wheel/rail contact model Fig. 1. Random dynamic analysis model of a train-track-bridge system.

122

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

(a)

The wheel/rail contact point area

Probability density function (PDF) of a random wheel/rail contact point on the rail surface

Wheel

PDF

mm

Rail

Xr

mm

(b) Time-dependent probability density evolution of the wheel/rail contact area

The center line of the rail Position of the train running V Fig. 2. The general view of the PDFs of a random wheel/rail contact point.

As shown in Fig. 2, the time-history PDFs of the wheel/rail contact points clearly reveal the physical significance of the wheel/rail contact interaction. The following assumptions are made for modeling the wheel/rail interaction: (1) Separation in the wheel/rail interaction and wheel derailment are neglected. (2) To easily find the wheel/rail contact point, the wheel/rail profiled tread is first assumed to be a rigid body. However, after the location is determined, the wheel/rail surface on the contact point is assumed to be an elastomer that is homogeneous and isotropic for calculating the wheel/rail contact force. (3) The contact area on the wheel/rail surface is assumed to be an elliptical region with much smaller dimensions than each body and much smaller than the relative radii of the wheel/ rail surface curvature. (4) The friction coefficient of the wheel/rail contact point is sufficiently high to ensure that no sliding occurs in the wheel/ rail interaction. (5) The bodies of the wheel/rail interaction are assumed to be quasi-identical, and the rolling direction is considered one of the principal axles of the assumed elliptical contact point. 2.1.2. Random wheel/rail interaction model As shown in Fig. 3, based on the assumptions in Section 2.1.1, a 3D wheel/rail interaction model is proposed, and the random track irregularity is considered the excitation. The 3D wheel/rail interaction model consists of the contact forces in the normal section and in the tangential planes, which are formulated using the Hertz theory [44] on the normal plane and the Kalker linear rolling contact theory [45] on the tangential plane. The Hertz theory requires the first three assumptions, and the last two assumptions support the Kalker theory. Assumptions (3) and (5) are illustrated in Fig. 3d. Random track irregularities are considered in the model (Fig. 3), where r ZL ðH; xk Þ and rZR ðH; xk Þ denote the vertical track irregularity and r YL ðH; xk Þ and rYR ðH; xk Þ denote the track alignment irregularity for the kth wheel-set. The subscripts ‘L’ and ‘R’ denote the left and right rails, respectively. The most important feature for modeling the wheel/rail interaction is the definition of the coordinate systems, which should be defined before establishing the model. First, as shown in

Fig. 3a and b, the absolute coordinate system O  XYZ is proposed for the train-track-bridge system, which is used to build the random dynamic equations. Second, each of the wheel-sets and the rails has their own local coordinate system, such as Ow  X w Y w Z w for the wheel-sets and Or  X r Y r Z r for each rail. The local coordinate systems, which change with the movement of the wheelset, are located at the center of the wheel-set or the rail under each wheel-set. Third, to obtain detailed descriptions of the wheel/rail contact points, each point has its own local coordinate system, e.g., the coordinate system of Oc  X c Y c Z c for contact point C is shown in Fig. 3d. Thus, to clearly understand the dynamic simulation of wheel/rail interaction, the local coordinate systems must be transformed into the absolute coordinate system. More details on this are provided in the following formula derivations. As shown in the 3D wheel/rail contact model in Fig. 3, d0 is half of the distance of the standard track gauge, o1 o2 is the centerline of the wheel-set, RwL and RwR are the instantaneous rolling radii of the left and right wheel, respectively, points B and B0 are the centers of the instantaneous rolling circle, and aL and aR are the intersection angles between the wheel/rail interface and the centerline o1 o2 . For each wheel-set, five degrees of freedom (DoFs) are considered for the kth wheel-set (k = 1,2,3,4): the longitudinal displacement xwk , the lateral displacement ywk , the vertical displacement zwk ; the roll displacement hwk , and the yaw displacement wwk . For the rail coordinates, taking the right rail as an example, six DoFs are considered: the longitudinal displacement xrR , the lateral displacement yrR , the vertical displacement zrR , the roll displacement hrR , the yaw displacement wrR , and the roll displacement urR . The coordinates for the left rail can be easily obtained by replacing R with L. The normal contact forces of the contact point C are F rL for the left rail and F rR for the right rail, which intersect with the centerline o1 o2 at A and A0 . The equivalent contact stiffness coefficient krail between the wheel and rail is linearized using the Hertz nonlinear contact theory (see Fig. 4) [10], the equation for which is F r ¼ ½G1 Dzwr  The formula is shown as

krail

   dF r  1:5 0:5  ¼ ¼ 1:5G Dzwr   dDzwr Dzwr ¼Dzwr;0

1:5

.

ð2Þ Dzwr ¼Dzwr;0

where G is the wheel/rail contact constant (m/N2/3), the wheel is worn type (LM), G = 4.57R0.149  108(m/N2/3), and R is the wheel

123

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

d0

(a)

d0

FrL o1 RwL

r ( r ZL (

xk )

wk

MzLcr

C

xk )

FyLcr rR

Random track irregularity of left rail

B C

A

zr R Z

O

o2 RwR

MzRcr

rail

wk

(b)

MzLcr

cr FyR

C'

C YR

r (

xk )

r ZR (

xk )

- FyLcr

Random track irregularity of right rail

X

FrL

RwL

R

yrR

rR

wk

' A' B

xrR

rR

rail

o1

ywk

B A

FrR

xwk

wk

B A

L

YL

zwk

(c)

-FrL

FyLcr - MzLcr

Y C' A' B'

o2

(d) FrL

xw k o1

B A C

wk

Xw Ow

ywk

C' A' B'

OrL

o2

ZC XC

OC

Yw

YC

XrR

XrL OrR

YrL

MzLcr

Creep areas

YrR

FxLcr

C

a

b

FyLcr

The coordinate system of the wheel/rail contact point C on left rail surface.

Fig. 3. 3D random wheel/rail contact model: (a) front view (in plane y-z); (b) top view (in plane x-y); (c) front view of left wheel/rail interaction; (d) Elliptical contact spot at location of C.

wheel-set using the absolute coordinate system and the local coordinate systems, as shown in Fig. 3. Several methods have been proposed to define the contact point C; the main methods are (a) the TLM [23,46], which finds the contact point via the wheel/rail contact geometry parameters, and (b) the equidistance iteration method [46], which involves more complex calculations. For convenience and practicability, TLM is employed in this paper to calculate the wheel/rail spatial geometric relationship. Based on Ref. [46], taking the wheel/rail contact point C in the left wheel-set as an example, the coordinates of the contact point C can be written as

Fr -1

Fr=[G

Fr0

-1.5

zw-r]

krail=tan

zw-r,0

zw-r

Fig. 4. The equivalent contact stiffness of the wheel/rail normal contact force on the wheel/rail surface.

nominal radius (m). F r is the wheel/rail normal force; Dzwr is the normal compression distance at the wheel/rail contact point (m); and Dzwr;0 is the initial compression distance under the static wheel axle load F r0 : The tangential contact forces, which are the creep forces transcr mitted between the wheel-sets and rails, are denoted F cr xL , F yL , and cr cr cr Mcr zL for the left wheel-set and F xR , F yR , and M zR for the right wheel-set. The superscript ‘cr’ denotes the tangential contact force, and the subscripts ‘L’ and ‘R’ denote the left and right rails, respectively. The most important step for the wheel/rail dynamic simulation is the location definition of the wheel/rail contact point C on each

8 xc ¼ xB þ lx RwkiL tan aL > >  > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > < y ¼ y  RwkiL l2 l tan a  l 1  l2 ð1 þ tan2 a Þ y L z L 2 c B x x 1lx >  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q > > > 2 2 RwkiL > lx lz tan aL þ ly 1  lx ð1 þ tan2 aL Þ : zc ¼ zB  1l 2

ð3Þ

x

where lx ¼  cos hwk sin wwk , ly ¼ cos hwk cos wwk , and lz ¼ sin hwk (k = 1,2,3,4). xB , yB ; and zB denote the coordinates of the wheel-set rolling circle center B. The coordinates xC ; yC and zC in Eq. (3) cannot be obtained directly in the absolute coordinate system. The coordinates of the wheel’s rolling circle center B (xB ; yB ; zB ) are related to both the coordinates of contact point C and the wheel-set motions. Therefore, we must first obtain the relative location of contact point C in the local coordinate systems of the wheel-set Ow  X w Y w Z w and the rail Or  X r Y r Z r . Then, the instantaneous rolling radii RwkiL and RwkiR , which equal the instantaneous distance between point

124

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

Fig. 5. The slope of the wheel/rail profile in the local coordinate system.

B and the wheel-set center, are calculated using the geometric shape of the wheel/rail tread surface based on the location of point C at the kth wheel-set of the ith vehicle. As shown in Fig. 5, the slopes of the wheel and rail surfaces must be equal when they are in contact, and the location of the wheel/rail contact point C is jointly determined by the wheel/rail instantaneous geometric relationships and the track/bridge instantaneous motion. Moreover, the tread gradient curves of the wheel/ rail surface at time t change instantaneously with the responses, including random track irregularities, track deformation, and dynamic responses of vehicles and the track-bridge system. Therefore, by translating the location of point B in the local coordinate system to the absolute coordinate system, we can obtain the location of the coordinates xC , yC and zC using Eq. (3).

where the superscript ‘T’ denotes the transpose of the matrix Xv i ði ¼ 1; 2; . . . ; nv Þ and nv is the number of vehicles, which includes two motor cars in the front and rear and six trailer cars in the middle of the train. The detailed format of the displacement vectors is listed in Appendix A. The train mass matrix Mv in Eq. (1), with the order N v  N v ðN v ¼ 38  nv Þ, takes the form

Mv ¼ diag½ Mv 1

   Mv n v 

ð5Þ

where the submatrix for each vehicle is

Mv i ¼ diag½ Mci

Mt1i

Mt2i

Mw1i

Mw2i

The train stiffness matrix Kv , N v  N v ðN v ¼ 38  nv Þ, can be written as

Kv ¼ diag½ Kv 1

2.2. The vehicle model

Mv 2

Kv 2

Mw3i

Mw4i 

with

ð6Þ

the

   Kv nv 

order

ð7Þ

where the submatrix for each vehicle is Similar to the vehicle models presented in many previous publications [21,34,47], the high-speed train in this study is composed of two motor cars in the front and rear and several trailer cars in the middle, which are all moving at a constant speed. Each vehicle is considered a complex multi-DoF dynamic system consisting of one car body, two bogies, and four wheel-sets, with several suspension spring-dampers. The vehicle model is established as a 4-axle vehicle with 38 DoFs, as shown in Fig. 6. The connections between the car body and bogies are represented by linear springs and viscous dashpots between two bogies and wheel-sets. The car body, bogies, and wheel-sets are regarded as rigid components and do not consider the randomness of the vehicle parameters. Thus, the wheel/rail contact area on the wheel-set should satisfy the assumptions in Section 2.1.1. Each car body and bogie has six DoFs: the longitudinal displacement xci and xtji , the lateral displacement yci and ytji , the vertical displacement zci and ztji , the roll displacement hci and htji , the yaw displacement wci and wtji , and the pitch displacement uci and utji , respectively. Each wheel-set has five DoFs: the longitudinal displacement xwki , the lateral displacement ywki , the vertical displacement zwki , the roll displacement hwki , and the yaw displacement wwki , with j = 1, 2 and k = 1,2,3,4 for the ith vehicle. Thus, the train displacement vector Xv in Eq. (1), with the order N v  1ðN v ¼ 38  nv Þ, can be written as

 Xv ¼ Xv 1

Xv 2

Xv 3

   Xv ðnv 1Þ

X v nv

T

ð4Þ

2 6 6 6 6 6 6 Kv i ¼ 6 6 6 6 6 4

Kci

Kct1i

Kct2i

0

0

0

Kt1i

0 Kt2i

Kt1w1i 0

Kt1w2i 0

0 Kt2w3i

Kw1i

0

0

Kw2i

0 Kw3i

syms

0

3

7 7 7 Kt2w4i 7 7 7 0 7 7 0 7 7 7 0 5 Kw4i 0

ð8Þ

where Kci , Ktji and Kwk i denote the stiffness submatrices for the car body, bogies, and wheel-set, respectively. Kctji and Ktjwki denote the matrices for the components including the ith car body, the jth bogie and the kth wheel-set (i ¼ 1; 2; . . . ; nv ; j ¼ 1; 2; k ¼ 1; 2; 3; 4). The detailed formats of the stiffness matrices are listed in Appendix A. The train damping matrix Cv in Eq. (1), with the order N v  N v ðN v ¼ 38  nv Þ; can be obtained by replacing k with c in the corresponding stiffness matrix Kv . The random load vector of the vehicles Fv ðH; xÞ in Eq. (1), with the order N v  1, involving the effect of random track irregularities, can be expressed as T Fv ðH; xÞ ¼ ½ Fv 1 ðH; xÞ Fv 2 ðH; xÞ    Fv nv ðH; xÞ 

ð9Þ

where Fv i ðH; xÞ denotes the random load vector for each vehicle; the detailed expression for the vector can be written as

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

125

Fig. 6. 3D vehicle model (a) left side view; (b) top view; (c) front view.

 Fv i ðH; xÞ ¼

016

f t1i ðH; xÞ

f t2i ðH; xÞ

f w1i ðH; xÞ

f w2i ðH; xÞ f w3i ðH; xÞ f w4i ðH; xÞ

T ð10Þ

In detail, the submatrices f tji ðH; xÞ in Eq. (10) denote the random load vector of the jth bogie, and the submatrices f wki ðH; xÞ denote the random load vector of the kth wheel-set. These submatrices can be expressed as

3T 0 7 6 0 7 6 7 6 2j 2j X X 7 6 ZL ZR ZL ZR 7 6 _ _ ðr ðH; x Þ þ r ðH; x ÞÞ þ c ð r ðH; x Þ þ r ðH; x ÞÞ k 1z 1z k k k k 7 6 7 6 k¼2j1 k¼2j1 7 6 7 6 2j 2j X X 7 f tji ðH; xÞ ¼ 6 ZL ZR ZL ZR 7 6 _ _ k1z b2 ðr ðH; xk Þ  r ðH; xk ÞÞ þ c1z b2 ðr ðH; xk Þ  r ðH; xk ÞÞ 7 6 7 6 k¼2j1 k¼2j1 7 6 7 6 2j 2j X X 7 6 kþ1 kþ1 ZL ZR ZL ZR 7 6 k1z Lt _ _ ð1Þ ðr ðH; x Þ þ r ðH; x ÞÞ þ c L ð1Þ ð r ðH; x Þ þ r ðH; x ÞÞ 1z t k k k k 7 6 5 4 k¼2j1 k¼2j1 2

ð11Þ

0

2R f wki ðH; xÞ ¼ f 1wki ðH; xÞ þ f 2L wki ðH; xÞ þ f wki ðH; xÞ

ð12Þ

f 1wki ðH; xÞ 2

3T 0 7 6 L R 22R _ YR _ YL 6 F rL kH;k ðxk Þ  F rR kH;k ðxk Þ þ 122L wki r ðH; xk Þ þ 1wki r ðH; xk Þ 7 7 6 7 ¼6 W axle 7 6 7 6 0 5 4 0 ð13Þ

126

(

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

Lx Ly Lz  f 2L f wki ðHÞ f wki ðHÞ M Lh M Lw wki ðH; xÞ ¼ ½ f wki ðHÞ wki ðHÞ wki ðHÞ

S

Rx Ry Rz f 2R  f wki ðHÞ f wki ðHÞ M Rh M Rw wki ðH; xÞ ¼ ½ f wki ðHÞ wki ðHÞ wki ðHÞ

ð14Þ

where aSwki and bwki (S = L, R) denote the half axis of the elliptical contact point between the kth wheel-set of the ith vehicle and the left and right rails at the corresponding positions, respectively; their

8 Lx cr f wki ðHÞ ¼ F cr > xL ðHÞ cos wwk þ F yL ðHÞ cosðaLk þ hwk Þsinwwk > > > > Ly cr > > f ðHÞ ¼ F cr xL ðHÞsinwwk þ F yL ðHÞ cosðaLk þ hwk Þcoswwk > < wki Lz cr f wki ðHÞ ¼ F yL ðHÞ sinðaLk þ hwk Þ cos wwk > > > cr cr cr > > M Lh > wki ðHÞ ¼ ðF xL ðHÞsinwwk  F yL ðHÞ cosðaLk þ hwk Þcoswwk ÞRwk L  F yL ðHÞ sinðaLk þ hwk Þ cos wwk d0 > > : Lw cr cr M wki ðHÞ ¼ ðF cr xL ðHÞ cos wwk þ F yL ðHÞ cosðaLk þ hwk Þsinwwk Þd0 þ M zL ðHÞ cosðaLk þ hwk Þ

ð15Þ

8 Rx cr f wki ðHÞ ¼ F cr > xR ðHÞ cos wwk þ F yR ðHÞ cosðaRk  hwk Þsinwwk > > > > Ry cr > > f ðHÞ ¼ F cr xR ðHÞsinwwk þ F yR ðHÞ cosðaRk  hwk Þcoswwk > < wki Rz cr f wki ðHÞ ¼ F yR ðHÞ sinðaRk  hwk Þ cos wwk > > > cr cr cr > > M Rh > wki ðHÞ ¼ ðF xR ðHÞsinwwk  F yR ðHÞ cosðaRk  hwk Þcoswwk ÞRwkR  F yR ðHÞ sinðaRk  hwk Þ cos wwk d0 > > : Rw cr cr M wki ðHÞ ¼ ðF cr xR ðHÞ cos wwk  F yR ðHÞ cosðaRk  hwk Þsinwwk Þd0 þ M zR ðHÞ cosðaRk  hwk Þ

ð16Þ

where H denotes the point set of random parameter vectors of track irregularities, r_ ðÞ is the first derivative of the track irregularity, the superscripts ‘Z’ and ‘Y’ denote the vertical and lateral directions, respectively, and ‘L’ and ‘R’ denote the left and right rails, respectively. kLH;k ðxk Þ and kRH;k ðxk Þ denote the wheel/rail contact angle slope of the left and right rails, respectively, at the kth wheel-set of the ith vehicle; here, kLH;k ðxk Þ ¼ tanðaLk þ hwk Þ, kRH;k ðxk Þ ¼ tanðaRk  hwk Þ. W axle is the axle weight involving the self-weight of all components of the vehicle distributed on each wheel-set; RwkL and RwkR denote the instantaneous rolling radius of the left and right sides of the wheels, respectively, at the kth wheel-set of the ith vehicle. cr cr The tangential contact forces F cr xL , F yL , and M zL in the left wheel/ cr cr rail pair and F cr xR , F yR , and M zR in the right wheel/rail pair can be calculated using the Kalker linear rolling contact theory (for more details, refer to Ref. [46]). Concretely, the tangential contact forces of the wheel/rail interaction can be written as

8 cr 11L L > < F xL ðHÞ ¼ 1wki -xwki ðHÞ cr L 23L L F yL ðHÞ ¼ 122L wki -ywki ðHÞ  1wki -wwki ðHÞ ; > : cr L 33L L M zL ðHÞ ¼ 123L wki -ywki ðHÞ  1wki -wwki ðHÞ

-Lwwki ðHÞ denote the longitudinal creep rate, the lateral creep

rate, and the spin creep rate of the wheel/rail contact point of the kth wheel-set on the ith vehicle acting on the left rail at the corresponding position, respectively; the corresponding values for the right rail are -Rxwki ðHÞ, -Rywki ðHÞ and -Rwwki ðHÞ, respectively. The creep coefficients obtained by [46] 8

mR 1mL wki and 1wki (m = 11, 22, 23, 33) can be

L Þ1 aLwki bwki C 11 L Þ1 aLwki bwki C 22

> 1 m > > > > < 122L ¼ 0:5E ð1þ m r wki ; > 123L ¼ 0:5Er ð1þ mÞ1 aL bL C 23 > > wki wki wki > > : 33L 1wki ¼ 0:5Er ð1þ mÞ1 aLwki bLwki C 33 11L wki ¼ 0:5Er ð1þ

8 R 1 R > 111R > wki ¼ 0:5Er ð1þ mÞ awki bwki C 11 > > > < 122R ¼ 0:5E ð1þ mÞ1 aR bR C wki

-Rxwki ðHÞ, -Rywki ðHÞ and -Rwwki ðHÞ, can be written as "

-Lxwki ðHÞ -Lywki ðHÞ

#T

2

x_ wki þ b0 w_ wki  x_ rL

3T

6 7 6 _ wki  y_ rL  h_ rL  r_ YL ðH; xk Þ 7 ¼ V 1 cL 4 y 5 _zwki þ b0 h_ wki  z_ rL  r_ ZL ðH; xk Þ 2 3 cos wwk sin wwk 6 7 6  cosðaLk þ hwk Þsinww  cosðaLk þ hwk Þ cos wwk 7 k 4 5 sinðaLk þ hwk Þ sin wwk  sinðaLk þ hwk Þ cos wwk

8 cr 11R R > < F xR ðHÞ ¼ 1wki -xwki ðHÞ cr R 23R R F yR ðHÞ ¼ 122R wki -ywki ðHÞ  1wki -wwki ðHÞ > : cr R 33R R M zR ðHÞ ¼ 123R wki -ywki ðHÞ  1wki -wwki ðHÞ

mR where 1mL wki and 1wki (m = 11, 22, 23, 33) denote the creep coefficient between the kth wheel-set and the left and right rails, respectively, at the corresponding position of the ith vehicle. -Lxwki ðHÞ, -Lywki ðHÞ,

and

S

product is aSwki bwki ¼ ae be ðF rkS RwkS Þ2=3 (S = L, R), where ae , be , and C m (m = 11, 22, 23, 33) are the coefficients obtained in Ref. [7]. Er denotes the elastic modulus of the rail, and m denotes the Poisson’s ratio. The creep rates of the wheel/rail contact point on the left rail, -Lxwki ðHÞ, -Lywki ðHÞ and -Lwwki ðHÞ, and those on the right rail,

r

wki wki 22 R Þ1 aRwki bwki C 23 R Þ1 aRwki bwki C 33

> > 123R > wki ¼ 0:5Er ð1þ m > > : 33R 1wki ¼ 0:5Er ð1þ m

ð18Þ

"

R xwki ðHÞ

-Rywki ðHÞ

#T

ð19Þ

ð17Þ

2

x_ wki  b0 w_ wki  x_ rR

3T

6 7 6 _ wki  y_ rR  h_ rR  r_ YR ðH; xk Þ 7 ¼ V 1 cR 4 y 5 ZR _ z_ wki þ b0 hwki  z_ rR  r_ ðH; xk Þ 3 2 cos wwk sin wwk 7 6 6  cosðaRk  hwk Þsinwwk  cosðaRk  hwk Þ cos wwk 7 5 4 sinðaR  hwk Þ sin wwk

ð20Þ

 sinðaRk  hwk Þ cos wwk

-Lwwki ¼ w_ wki cosðaLk þ hwk Þ; -Rwwki ¼ w_ wki cosðaRk  hwk Þ

ð21Þ

where V cL and V cR denote the nominal speeds of the wheel/rail contact point on the left and right rails, respectively, which can be obtained by

V cL ¼ 0:5v c ð1 þ RwkL R1 0 coswwk Þ; V cR ¼ 0:5v c ð1 þ RwkR R1 0 coswwk Þ

ð22Þ

127

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

where v c denotes the train speed, RwkL and RwkR denote the instantaneous rolling radius of the left and right wheel-sets, respectively, and R0 is the nominal rolling radius of the wheel-set. 2.3. The rail model and wheel/rail interaction As shown in Eq. (2), the rail matrices are indicated by the subscript ‘r’. The steel rails, composed of the left and right rails, are modeled as uniform Bernoulli-Euler beams composed of two nodes, each of which has six DoFs. For ease of expression, UikX , UYik , UikZ and Uwik , with the order 1  N rL ðN rL ¼ 6  nr Þ, denote the shape functions for the rail element in the longitudinal direction X, the lateral direction Y, the vertical direction Z and the rotation direction w, respectively, evaluated at the position of the kth wheel-set of the ith vehicle.Um ik (m = X,Y,Z,w) is a time-varying process due to the vehicle running on each rail element. The details are provided in Appendix A. The rail matrices Mr , Kr and Cr , with the order N r  N r ðN r ¼ 2  6  nr Þ, can be written as

M r ¼ diagf M rL Kr ¼ diagf KrL þ

M rR g; nv X 4 X

Cr ¼ diagf CrL

T krail ðUikZ Þ UikZ

CrR g

ð23Þ

KLv ikr

6 6 ¼6 4

KRv ikr ;

0ð38iþ5k18ÞNrL krail UikZ krail b0 UikZ 0ðNv 38i5kþ16ÞNrL

ð27Þ

FrL ðH; xÞ ¼ F1rL ðH; xÞ þ F2rL ðH; xÞ þ F3rL ðH; xÞ FrR ðH; xÞ ¼ F1rR ðH; xÞ þ F2rR ðH; xÞ þ F3rR ðH; xÞ

Krv ¼ ðKv r Þ

T

3

2

7 7 7 5

6 6 ¼6 4

;

KRv ikr

N v N rL

ð25Þ 0ð38iþ5k18ÞNrL krail UikZ krail b0 UikZ 0ðNv 38i5kþ16ÞNrL

3 7 7 7 5

8 nv X 4 X > Y T > _ YL > F1 ðH; xÞ ¼  ðF rL kLH;k ðxk ÞUYik þ 122L > wki r ðH; xk ÞUik Þ < rL i¼1 k¼1

nv X 4 > X > 1 > Y T > _ YR ðF rR kRH;k ðxk ÞUYik þ 122R : FrR ðH; xÞ ¼  wki r ðH; xk ÞUik Þ

N v N rL

where KLv ikr and KRv ikr denote the stiffness matrices for the wheel/rail interaction on the left and right rails, respectively. b0 is the halfdistance of the rail gauge in the ‘Y’ axis direction. The equivalent

ksby , csby

i¼1 k¼1

0 "

F2rL ðH; xÞ

#

" nv X 4 B X B f Lx wki B ¼ B Rx i¼1 k¼1 @ f wki

f wki

Ly

f wki

Lz

Ry

f wki

Rz

f wki

2 X 31T Uik # 6 Y 7C 7C 6 MLw wki 6 Uik 7C 6 U Z 7C Rw Mwki 4 ik 5A Uwik

8 nv X 4 X > 3 > > ðk1z r ZL ðH; xk ÞUikZ þ c1z r_ ZL ðH; xk ÞUikZ Þ > FrL ðH; xÞ ¼  < i¼1 k¼1

nv X 4 > X > 3 > > ðk1z r ZR ðH; xk ÞUikZ þ c1z r_ ZR ðH; xk ÞUikZ Þ : FrR ðH; xÞ ¼ 

ð30Þ

ð31Þ

i¼1 k¼1

F1rL ðH; xÞ

where and F1rR ðH; xÞ denote the lateral random load vector induced by the normal wheel/rail contact force and the lateral creep force induced by random alignment track irregularity, respectively; F2rL ðH; xÞ and F2rR ðH; xÞ denote the random load vectors induced by the wheel/rail tangential contact forces. F3rL ðH; xÞ and F3rR ðH; xÞ denote the random load vectors induced by random vertical profile track irregularities. The subscripts ‘L’ and ‘R’ denote the left and right rail, respectively.

The matrices for the track-bridge system are indicated by the subscript ‘tb’. Ballastless track systems, which are composed of ballastless track slabs and concrete base plates, are modeled as finite plate elements in the high-speed railway. The simply supported concrete bridge systems, which consist of bridge beams and piers, are modeled as 3D finite beam elements.

krsy ,crsy

k sbz ,csbz

krsz ,crsz kbby ,cbby

Ballastless track slab

Bridge

ð29Þ

2.4. The track-bridge model

ð26Þ

Rail

ð28Þ

with

i¼1 k¼1

where M rL and M rR , with the order N rL  N rL ðN rL ¼ 6  nr Þ, denote the mass matrices of the left and right rails, respectively; KrL and KrR , with the order NrL  NrL , denote the stiffness matrices of the left and right rails, respectively; and CrL and CrR , with the order NrL  NrL , denote the damping matrices of the left and right rails, respectively. The matrices induced by the vehicle-rail interaction are denoted with the subscript ‘vr’ or ‘rv’. The stiffness matrix Kv r , with the order N v  N r , induced by the wheel/rail interaction can be written as

2

(

F2rR ðH; xÞ

ð24Þ

i¼1 k¼1

Fr ðH; xÞ ¼ ½ FrL ðH; xÞ FrR ðH; xÞ T

nv X 4 X T KrR þ krail ðUikZ Þ UikZ g

i¼1 k¼1

nv X 4 X Kv r ¼ ½ KLv ikr

contact damping coefficient of the wheel/rail interaction is neglected; therefore, the damping matrix Crv is neglected. The random load vector Fr ðH; xÞ of the rail in Eq. (2), which includes the direct effect of random track irregularities, with the order N r  1ðN r ¼ 2  6  nr Þ, can be written as

b

kbbz , cbbz

e

zb

xb kbpy ,cbpy kbpz ,cbpz

Pier

Fig. 7. Finite element model of the rail-track-bridge-pier system.

h

128

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

As shown in Fig. 7, the matrices of the track-bridge coupled system, Mtb , Ktb and Ctb , with the order ðN t þ N b Þ  ðN t þ N b Þ, can be easily obtained using the finite element method. The stiffness matrices (Krtb and Ktbr ) and the damping matrices (Crtb and Ctbr ) of the rail-track-bridge interaction, indicated by the subscripts ‘rtb’ or ‘tbr’, are induced by the spring-dampers between each assembly element. In detail, the matrices are such that

Crtb ¼ CTtbr

Krtb ¼ KTtbr ;

ð32Þ

Except for the random track irregularities, the influence of other random excitations acting on the train-track-bridge system are neglected. Thus, the load vector Ftb ðH; xÞ, with the order ðN t þ N b Þ  1, is zero.

A good point set with a 2N-dimensional hypercube [51], which is an effective method for defining point sets, is generated by the sequence



j

j

vq;j ¼ qH2Nþ1  int qH2Nþ1



ðq ¼ 1; 2; . . . ; npt ; ; j ¼ 1; 2; . . . ; 2NÞ ð35Þ

where H denotes a prime number, vq;j 2 ð0; 1Þ denotes the discrete point of a 2N-dimensional hypercube, and npt denotes the total number of the representative track irregularity samples. Thus, the representative discrete points can be obtained by [39]

(

ðpÞ 1

ðXq;i Þ1 ¼ ðXi1 Þ

ðpÞ 1

þ ½ðXi1 Þ

ðpÞ 1

 ðXi Þ vq;i

ð36Þ

/q;i ¼ 2pvq;iþN 3. Random dynamic analysis of a train-track-bridge system using PDEM 3.1. Generation of random track irregularity representative samples Track irregularity is considered one of the major factors causing random vibration of train-track-bridge systems. Many traditional methods, such as the trigonometric series method [48] and the second-order filtering method [49], are used to generate track irregularity samples. Each of these methods has its own special applications, which may entail certain limitations. Recently, SHF combined with PDEM [37] was proposed as a new approach for generating representative random track irregularity samples with probability characteristics [39], which clearly improved the computational efficiency. Based on PDEM, the system random dynamic characteristics can be revealed. As mentioned previously, the symbol H ¼ ðn1 ; n2 ; . . . ; nnpt Þ represents the random variable space of track irregularities. The vertical track profile irregularity of the left rail r ZL ðH; xÞ and the right rail rZR ðH; xÞ and the track alignment irregularity of the left rail rYL ðH; xÞ and the right rail rYR ðH; xÞ, which are shown separately in Fig. 3, can be obtained for the actual track geometry using

8 ZL r ðH; xÞ ¼ cz ðH; xÞ þ ch ðH; xÞb0 > > > > < r ZR ðH; xÞ ¼ c ðH; xÞ  c ðH; xÞb0 z h > r YL ðH; xÞ ¼ cy ðH; xÞ þ ct ðH; xÞ=2 > > > : r YR ðH; xÞ ¼ c ðH; xÞ  c ðH; xÞ=2 y

ð33Þ

N X AI ðXIH;i Þ cosðXIH;i x þ /IH;i Þ I ¼ z; h; y; t

ð34Þ

i¼1

where XIH;i 2 H and /IH;i 2 H (i ¼ 1; 2; . . . ; N; I ¼ z; h; y; t) denote the random spatial frequencies and random phases of the track spectrum, respectively, and N is the number of random variables. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AI ðXIH;i Þ ¼ 2SI ðXIH;i ÞDXIH;i (I ¼ z; h; y; t) is an amplitude function related to the random frequencies XIH;i , and SI ðXIH;i Þ is the power spectrum. The key point for generating cI ðH; xÞ is the definition of the representative discrete point sets of the random spatial frequencies

XIH;i and random phases /IH;i . Therefore, a strategy for defining the representative point sets using NTM [51,52] is proposed.

nq ¼ fXq;1 ; Xq;2 ; . . . ; Xq;N ; /q;1 ; /q;2 ; . . . ; /q;N g 2 H q ¼ 1; 2; . . . ; npt

ð37Þ

Substituting Eq. (36) into Eq. (34) yields

cI ðH; xÞ ¼

N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2SI ðXIH;i ÞDXIH;i cosðXIH;i x þ /IH;i Þ I ¼ z; h; y; t

ð38Þ

i¼1

Here, cI ðH; xÞ presents the representative track irregularity sample generated by SHF. 3.2. Solution with PDEM As stated in Section 2, the train-track-bridge system is assumed to be a conservative system, which means that the only source of randomness comes from the random track irregularities and that the randomness of the system parameters is neglected. The total inflow probability at any domain of the state space is equal to the outflow probability transited through the boundary of the domain [36]. For convenient expression, the spatial domain ‘x’ in Eq. (1) is transformed into the time domain ‘t’, as x = vt, where v is the train speed. Thus, the dynamic equation of train-track-bridge systems in Eq. (1) can be rewritten as

€ þ CðtÞX_ þ KðtÞX ¼ FðH; tÞ MX

t

where cz ðH; xÞ denotes the random vertical track profile irregularity sample, ch ðH; xÞ denotes the random track cross-level irregularity sample, cy ðH; xÞ denotes the random track alignment irregularity sample and ct ðH; xÞ denotes the random track-distance irregularity sample. b0 is the half-distance of the track width. Similar to the theories mentioned in Ref. [39,50], SHF is employed to generate random track irregularity samples with track spectrum, which is shown as

cI ðH; xÞ ¼

where i ¼ 1; 2; . . . ; N. Thus,

ð39Þ

First, one can obtain the representative discrete points set nq ¼ fXq;1 ; Xq;2 ;    ; Xq;N ; /q;1 ; /q;2 ; . . . ; /q;N g 2 H in the random variable space H, where q ¼ 1; 2; . . . ; npt . The discrete point sets of representative random spatial frequencies XIH;i 2 nq and random phases /IH;i 2 nq (i ¼ 1; 2; . . . ; N; I ¼ z; h; y; t) are filtered with NTM using Eqs. (36) and (37). The initial probability of each variable point set is denoted as P q , and the total probability of the train-track-bridge system satisfies

Z Z npt npt Z X X Pq ¼ pH ðnq Þdn ¼ [ pH ðnq Þdn ¼ npt q¼1

q¼1

nq

nq

XH

pH ðnq Þdn ¼ 1

q¼1

ð40Þ The initial conditions are correspondingly partially discretized as

  pUH ðu; nq ; tÞt¼t ¼ dðu  u0 ÞpH ðnq ; tÞt¼t ¼ dðu  u0 ÞPq 0

0

ð41Þ

where dðÞ is a Dirac delta function. Second, using the Newmark-b time integral method, the dynamic solution of train-track-bridge motion in Eq. (39) can be

129

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

Fig. 8. The wheel-set with LM type tread and the steel rail (Units: mm).

Random track irregularites

speed v

1

2

m

m 1:20

m

2

1:20

m

1

Ac =7.60 m2 I y =13.45 m4 I x =35.17 m4 I t =245.53 G·Pa·m4 Ec =3.15×10 4 N/mm2

m

m Precast track slab. d=200 mm

Vehicles

Mortar adjustment layers. d=30 mm Concrete bed plate. d=200 mm

Subgrade

Pier

Ac =8.89 m2 I y =10.95 m4 I z =85.90 m4 I t =289.8 G·Pa·m4 Ec =3.45×10 4 N/mm2 Mid section 2-2

Ac =12.95 m2 I y =15.23 m4 I z =99.45 m4 I t =345.6 G·Pa·m4 Ec =3.45×10 4 N/mm2 End section 1-1

Fig. 9. 3D train-track-bridge model for a high-speed railway.

Table 1 The natural frequencies and modes of the bridge. No.

Freq./Hz

Modes

1 2 3 4 5

2.731 3.711 4.061 5.026 5.139

Pier/beam longitudinal bending Beam, lateral symmetric Beam, lateral asymmetric Beam, lateral symmetric Beam, longitudinal bending

_ and X. € The solution X is directly influenced by obtained, i.e., X, X the randomness of nq 2 H.

Without loss of generality, take U as the state vector that denotes the random response of the train-track-bridge system, which can be written as

_ X; € n ; tÞ; U ¼ GðX; X; q

_ X; € n ; tÞ ðl ¼ 1; 2; . . . ; nd Þ U l ¼ Gl ðX; X; q ð42Þ

where U ¼ ðU 1 ; U 2 ; . . . ; U nd ÞT , GðÞ denotes the transfer function of the dynamic response, and nd denotes the dimension of the transfer _ also depends on n ¼ H and can function U. Similarly, the velocity U q

be written as

130

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

_ X; € n ; tÞ; U_ ¼ gðX; X; q

_ X; € n ; tÞ ðl ¼ 1; 2; . . . ; nd Þ U_ l ¼ g l ðX; X; q ð43Þ

where gðÞ denotes the transfer function of the velocity of the dynamic response. According to Eq. (42) and Eq. (43), gðÞ ¼ @GðÞ=@t. The conditional PDF of U is

_ X; € n ; tÞÞ pUjH ðu; tjnq Þ ¼ dðu  GðX; X; q

ð44Þ

Therefore, based on Eq. (43), the joint probability density evolu_ X; € n ; tÞ yields tion of ðX; X; q

_ X; € n ; tÞÞp ðn Þ pUH ðu; nq ; tÞ ¼ dðu  GðX; X; q H q

form. The PDEM calculation results are verified by comparison with the results obtained using MCM. The track irregularity used here is normally considered a nonstationary function due to the subgrade/bridge transition section, which means that the amplitudes of the track irregularity samples on the bridge are c times smaller than those on the subgrade [31]. Therefore, a slowly varying sinusoidal modulation function [39] is employed to generate non-stationary track irregularity samples, with a modulation rate of 0.7. The calculation begins when the train is located at -100 meters away from the bridge, ensuring that the vehicle responses are stable when the train reaches the bridge.

ð45Þ

Third, based on the Reynolds transport theorem and its related derivation, by differentiating Eq. (45) on both sides with regard to t [38], we obtain _ X; € n ; tÞÞp ðn Þ @pUH ðu; nq ; tÞ @½dðu  GðX; X; q H q ¼ @t @t _ X; € n ; tÞÞp ðn Þ @ðu  GðX; X; _ X; € n ; tÞÞ @½dðu  GðX; X; q q H q ¼ @u @t _ X; € n ; tÞ @p ðu; n ; tÞ @GðX; X; q q UH ¼  @t @u @pUH ðu; nq ; tÞ _ ¼ Uðn q ; tÞ  @u ð46Þ Rearranging Eq. (46), the generalized probability density evolution equation [38] is obtained as

4.2. Verification of PDEM by comparison with MCM using dynamic analysis As the upgraded version of the train-bridge model established by Yu and Mao et al. [39], the train-track-bridge model with refined 3D wheel/rail contact interaction in the present paper is much more complex and more realistic than the previous model. MCM is employed to verify the reliability of PDEM in the present study. Random track irregularity is considered to be the only excitation for the train-track-bridge system. The time-history curves of the random dynamic response of the bridge and vehicles are considered to be the typical objects of the train-bridge system. To reduce the calculation time and to improve the computational efficiency, in this section, the bridge model is reduced to a single simply supported beam; the other conditions remain the same.

ð49Þ

4.2.1. PDF of the random response The 3D time-varying process PDF of random vertical displacement at the center of the third vehicle body and its probability density contours are shown in Figs. 10 and 11, respectively. The train speed is 250 km/h. The 3D time-varying process PDF, resembling a mountain stretching into the distance, vividly shows the physical characteristics of the vertical vehicle displacement. The whirlpools in the contour in Fig. 11 are related to the peaks and valleys of the ‘‘mountain”. This figure clearly shows the random dynamic characteristics of the train-bridge coupled interaction from the PDF at different times. The PDF in Fig. 10 implies that the train-bridge coupled interaction is a complex random evolutionary process quite different from a stationary process.

Based on the refined wheel/rail contact model proposed in this paper, a 3D train-track-bridge dynamic model for a high-speed railway involving random track irregularity is investigated. Taking the abradability of the wheel/rail contact as an example, a wheelset with LM type tread and a steel rail of 60 kg/m are used as an example in the simulation (Fig. 8). A bridge model featuring three simply supported prestressed concrete box girders with spans of 32 m each is established, and the train model contains eight vehicles, with two motors at the front and rear and six trailer motors in the middle (Fig. 9). The major parameters of the vehicle are listed in Table 2 in Appendix B. The natural frequencies and the bridge modes are listed in Table 1. Rayleigh damping is employed to form the bridge damping matrix, and the damping ratio is 0.05. The track irregularity samples, which are generally regarded as random excitation, are generated by a German low-interference track spectrum [53], whose cut-off spatial frequency ranges from 0.04 rad/m to 3.14 rad/m. MATLAB@ software is used as the computing plat-

4.2.2. Verification of PDEM with MCM Due to the random track irregularities, which are considered to be one of the main random factors, the train-bridge system response presents strong randomness. As mentioned previously, MCM was used to verify PDEM due to its efficiency at an acceptable calculation precision. According to the curves in Fig. 12, the results obtained from PDEM matched well with those obtained from MCM; the former was derived from 400 representative samples, and the latter was derived from 9999 samples. The maximum deviation in the vehicle response obtained by PDEM and MCM is less than 1.84%. On the same computer, the time required for the PDEM calculation was 31 h, whereas that for MCM was 680 h, showing an improvement of 1–2 orders of magnitude. This proves that PDEM has a higher efficiency than MCM at the same calculation accuracy. As shown in Fig. 12a and Fig. 12b, the peak values of PDF from 1.0 s to 2.0 s are significantly higher than the values at other times, similar to the amplitude trend of the mean value of vehicle displacement in Fig. 12c. This result implies that the train runs more stably on the bridge than on the embankment, which is due to the nonstationary track irregularities from the embankment to the bridge. Normally, the random dynamic characteristics of the vehicle responses under train-bridge coupled interaction cannot be revealed very clearly by a single deterministic time-history curve;

@pUH ðu; nq ; tÞ _ @pUH ðu; nq ; tÞ þ Uðnq ; tÞ ¼0 @t @u

ð47Þ

where the initial condition yields

 pUH ðu; nq ; tÞt¼0 ¼ dðu  u0 ÞpH ðnq Þ

ð48Þ

Finally, the generalized probability density evolution equation of train-track-bridge system in Eq. (47) is solved using the bilateral difference method with the functionality of a total variation diminishing format [37] to obtain the probabilistic solution pUH ðu; nq ; tÞ: The PDF of the dynamic response of a train-track-bridge system is given as

pU ðu; tÞ ¼ ¼

R

RUH UH

_ X; € n ; tÞÞp ðn Þdn dðu  GðX; X; q H q pUH ðu; nq ; tÞdn

For more details on PDEM, refer to Ref. [37]. 4. Numerical examples and validation 4.1. Numerical model information

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

131

Fig. 10. PDF of random vertical displacement of the third vehicle body at the center (v = 250 km/h).

Fig. 11. The probability density contours of random vertical displacement of the third vehicle body at the center on Fig. 10 (v = 250 km/h).

however, the statistical data from a large number of results would be more convincing. The mean value curve in Fig. 12c shows that the curve decreases at the first and then increases from 1.0 s to 2.0 s. The initial decrease is caused by the bridge deflection due to the vehicle weight when the train runs across the bridge. The standard deviation curve in Fig. 12d also decreases due to the modulation of the nonstationary track irregularity samples, indicating that the amplitude of the track irregularity is generally small on the bridge than on the embankment. Therefore, the standard deviation of the vertical car body displacement is smaller on the bridge than on the embankment. The ratio of the minimum to maximum values in Fig. 12d is nearly 0.75, which is close to the modulation rate of 0.7. In addition to the random vehicle displacement shown in Fig. 12, the other three major random dynamic responses are shown in Fig. 13. These figures show that the results obtained from both PDEM and MCM matched well, even though the former was derived from 400 representative samples and the latter was derived from 9999 samples. This result verifies that PDEM is more efficient than MCM at the same calculation accuracy, representing an efficiency improvement of 1–2 orders of magnitude.

4.3. Random dynamic analysis of wheel/rail interaction The simulation of the wheel/rail interaction is considered one of the key points for safety assessments of train-bridge coupled system. Deeply understanding the dynamic characteristics of wheel/ rail interaction will result in better design and safer operation of high-speed railways. After verifying PDEM with MCM in Section 4.2, the numerical model of the train-track-bridge system established in Section 4.1 is demonstrated to be available for further simulation. In the traditional deterministic analytical method, the wheel/ rail contact is generally assumed to be a single-point contact, and the wheel/rail contact point on the surface was obtained as a single time-varying trace line. In reality, the wheel/rail contact point located on the wheel/rail surface is randomly distributed within a range. This type of randomness could be described with a certain time-varying probability density evolution function. Thus, the contours of the probability density evolution functions on the random wheel/rail contact point C at the first wheel-set on the left rail are shown in Fig. 14, which is taken as a numerical example.

132

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

Fig. 12. PDFs of the random vertical displacement process that varies with time at the center of gravity of the third car body and the verification of these results with MCM: (a) 3D PDF; (b) Contour of the PDF; (c) Mean value curves; (d) Standard deviation curves (v = 250 km/h).

Fig. 13. Verification of PDEM with MCM for the random dynamic responses: (a) and (b) the mean and standard deviation of the bridge deflection at the mid-span; (c) and (d) the mean and standard deviation of the bridge acceleration at the mid-span; (e) and (f) the mean and standard deviation of the vehicle displacement at the center of the car body (v = 250 km/h).

As depicted in Fig. 14b, the width of the rail cross section is 72 mm, and the calibration at the rail centerline is 36 mm. Due to the wheel/rail interaction caused by train-bridge coupling, the PDF contours of the wheel/rail contact point C cannot be symmetrically distributed on both side of the centerline when the train runs across the bridge. In reality, the random dynamic responses of the subgrade strongly influence the location of the wheel/rail contact point C on the rail surface. Taking the subgrade-bridge transition section as an example, as shown in the enlarged figure in Fig. 14c, the PDF contours of the wheel/rail contact point C smoothly changed at 1.20 s. One of major reasons for this result is the transition features of the subgrade and bridge, where the substructure stiffness is different for the subgrade and bridge. As shown in Fig. 14a, the PDF contours of the wheel/rail contact point C on the rail surface from 1.4 s to 2.7 s are much more centralized near the centerline than those of other time intervals with

a smaller range of the probability function. In addition to the influence of the train-bridge coupled interaction, the non-stationarity of the track irregularity samples, which have smaller amplitudes on the bridge than on the subgrade, are considered to be another major reason for this result. Specifically, the range of PDF contours on the bridge does not immediately decrease at the first span of the bridge mainly because the random dynamic responses of the wheel-set need some time to reach a steady state after the train moves across the subgrade-bridge transition. To better understand the probability density evolution function of the wheel/rail interaction, several representative sections of the PDF contours in Fig. 14a are shown in Fig. 15 and are respectively marked as A to E. The probability ranges and their PDFs in Fig. 15 present the probability of the wheel/rail contact point C appearing on the rail surface. The figure clearly shows that the PDFs differ significantly in different sections. The PDFs at the center of gravity for

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

133

Fig. 14. The probability density evolution function of a random wheel/rail contact point at the first wheel-set of the vehicle on the left rail surface.

Fig. 15. PDF of the wheel/rail contact point C in Fig. 14a at different typical times.

the three spans of the bridge are quite different from those on the subgrade. As a result, combining the PDEM and the random wheel/rail contact model, the PDF contours clearly show the distribution range and the probability density values of the wheel/rail contact point. The wheel/rail contact point is actually randomly distributed on the rail surface, which is not represented by the deterministic values obtained by the traditional method. This conclusion explains why there is always a polished stripe on the centerline of each rail surface. 4.4. Comparison of the Chinese track spectrum and the German track spectrum The track spectrum is one of the most important parts of the train-bridge dynamic analysis. The running safety and reliability assessment of a high-speed railway depend on the smoothness of the track irregularity samples that are obtained from field measurements or generation. There are several typical track spectra

for the dynamic simulation of high-speed railways, including the German low-interference track spectrum, Japanese track spectrum, and the newly developed Chinese high-speed railway track spectrum. To provide a thorough understanding of the application scope of each track spectrum in the train-bridge dynamic calculation, the track spectra must be compared. Because the German track spectrum is much more wildly employed for dynamic simulations than the Japan track spectrum for railways, the dynamic calculations here are based on the German track spectrum and the Chinese high-speed railway track spectrum. The train speed is 250 km/h. There are many important indictors that considered references for train-bridge safety assessments, such as the bridge displacement, bridge deck accelerations, vehicle response, and derailment. Due to the limitations on this paper, this study mainly focuses on the bridge displacement and vehicle acceleration. Under the excitation of random track irregularities, the timevarying PDF contours of the bridge deflection at the mid-span show similar trends in Fig. 16a and b. However, the partially

134

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

Fig. 16. Comparative analysis of random bridge deflections at the first mid-span using the German low-interference track spectrum and the Chinese high-speed railway track spectrum (v = 250 km/h).

Fig. 17. Comparative analysis of the vertical vehicle acceleration at the center of the fourth car body using the German low-interference track spectrum and the Chinese highspeed railway track spectrum (v = 250 km/h).

enlarged figures at A and B in Fig. 16 clearly show that the PDF contours have considerable differences, even though their mean values are approximately equal. In reality, the range of the PDF contour indicates the upper and lower limits of random bridge deflection. The closer a response is to the mean value curve, the higher its probability density value is. The mean values plus/minus three times the standard deviation values based on the normal distribution are used for comparison with the PDF contours; this comparison shows that most contours fall within the upper and lower limits. Furthermore, the upper and lower limits of random bridge deflection in Fig. A at 1.18 s are 0.974 mm and 1.107 mm, respectively, with Ddg = 0.133 mm, whereas in Fig. B, the upper and lower limits are 1.020 mm and 1.071 mm, respectively, with Ddc = 0.051 mm. The value of Ddg =Ddc is 2.60, which means that the German track spectrum show a greater effect of track irregularity on the bridge dynamic responses than the Chinese high-speed railway spectrum. This result directly demonstrates

that track irregularity in the Chinese high-speed railway spectrum has greater smoothness than that in the German track spectrum. A similar comparison of random vehicle responses is undertaken in Fig. 17 (vertical vehicle acceleration) and Fig. 18 (lateral vehicle acceleration). These figures clearly show that the amplitudes of the vehicle acceleration have considerable differences for the different track spectra. First, the mean value curves show that the vehicle accelerations have a mean value of approximately zero, except for the section in which the train runs across the bridge at approximately 1.8 s to 3.2 s due to the influence of the train-bridge coupled interaction. The contours of the PDF are symmetrically distributed on both sides of the mean value, and the amplitude of the vertical vehicle acceleration from the upper limit to the lower limit in Fig. 17a at 2.20 s to 2.30 s is 0.430 m/s2 to 0.422 m/s2 with Dazg = 0.852 m/s2, whereas in Fig. 17b, the amplitude of the upper limit to the lower limit is 0.140 m/s2 to -0.171 m/s2 withDazc = 0.311 m/s2. The value of

135

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

Fig. 18. Comparative analysis of the random lateral vehicle acceleration at the center of the fourth car body using the German low-interference track spectrum and the Chinese high-speed railway track spectrum (v = 250 km/h).

Dynamic coefficient

4.0

Calculated value ( f1=4.67Hz ), Ref. [46]

3.0

Calculated value ( f2=6.00Hz ), Ref. [46] 2.5

Measured values, Ref. [46] Calculated by PDEM ( f0=4.95Hz )

2.0 1.5 1.0 0

50

100

150

200

250

300

350

400

450

500

550

600

Train speed (km/h) Fig. 19. Random dynamic analysis of the maximum bridge response at the mid-span of a 32 m simply supported girder concrete bridge versus train speed.

Dazg =Dazc is 2.74, which is close to that obtained for the bridge deflection. Using the same method, the amplitude of the lateral vehicle acceleration from the upper limit to the lower limit in Fig. 18a at 2.30 s to 2.40 s is 0.315 m/s2 to 0.332 m/s2 with Dayg = 0.647 m/ s2, whereas in Fig. 18b, the amplitude of the upper limit to the lower limit is 0.092 m/s2 to -0.091 m/s2 with Dayc = 0.183 m/s2. In this case, the value of Dayg =Dayc is 3.54. The analysis results for the vehicle and bridge responses clearly shows that the track irregularity samples generated by the Chinese high-speed railway track spectrum are smoother than those generated by the German track spectrum and that the amplitudes of the dynamic response are two to three times greater for the latter than for the former.

4.5. Speed analysis Train speed is another important factor that influences the running safety assessment in high-speed railways. Fig. 19 illustrates the maximum dynamic coefficient of bridge deflection versus the train speed based on the comparison of the train-track-bridge model established in Section 4.1 using the Chinese high-speed railway track spectrum and the calculation results and field test

results published in Ref. [54]. The dynamic coefficient g is defined as

g¼

X dynamicv  100% X static

ð50Þ

where X dynamicv denotes the maximum bridge response at the midspan when the train runs across the bridge at speed v and X static denotes the static bridge response at the mid-span. The bridge resonances directly influence the vehicle running safety when the train runs at a high speed. When bridge resonance occurs, the dynamic coefficient of the bridge dynamic response increases sharply, affecting the high-speed railway running safety and passenger comfort. Thus, it is important to determine the resonance speed using the dynamic analysis to avoid bridge resonance. As shown in Fig. 19, the calculated dynamic coefficients of bridge deflection versus train speed match the measured values well. The dynamic coefficient curve obtained using PDEM has a peak value that falls in between the peak values of the other two curves. However, the PDEM curve has the same order of magnitude as the other two curves when the train speed is smaller. All of the curves of the dynamic coefficient versus train speed increase and reach their peak values at a certain resonance speed; before this speed, the dynamic coefficient remains small. The train speed at

136

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

(d) The dynamic coefficient of bridge deflection increases with increasing train speed before reaching the resonance speed, which is consistent with published results.

the bridge resonance continues to increase with increasing bridge natural frequency. Nearly all of the measured results are within the maximum limit values. The 32 m simply supported girder bridge has two distinct resonant velocities versus train speed. For example, the results in Fig. 19 for f 1 ¼ 4:67 Hz show the first peak at 140 km/h (38.9 m/ s) and the highest peak near 420 km/h (116.7 m/s). The bridge resonance phenomenon may be caused by the wheel load forced frequency being close to the natural frequency of the bridge at a particular speed. The wheel load forced frequency f b can be obtained using the formula f b ¼ v =L, where v denotes the train speed and L denotes the vehicle length (or the bogie length on the vehicle or near the vehicle). Additionally, taking the calculated curve in Fig. 19 for f 1 ¼ 4:67 Hz as an example, when the train speed reaches 420 km/h, the wheel load forced frequency is f b;v = 116.7 (m/s)/24.775 (m, the vehicle length) = 4.71 Hz, which is close to the natural frequency of the bridge f 1 ¼ 4:67 Hz. Thus, the bridge experiences dynamic resonance phenomenon, which follows the same principle as that for the first peak resonance. Overall, for random track irregularity, the case study in Fig. 19 verifies that the established train-track-bridge model that is calculated using PDEM is efficient and accurate and that the dynamic calculation using PDEM can clearly reveal the random dynamic mechanisms of a train-bridge coupled system.

Acknowledgements Joint support was received from the National Natural Science Foundation of China (Grant No. 51578549), the Joint Fund of the National Natural Science Foundation of China (Grant No. U1434204), and the science and technology research and development program of the China Railway Corporation (SY2016G001). Appendix A (1) Vehicle matrices The detailed format of the displacement vector for each vehicle in Eq. (3) is

Xv i ¼ ½Xci Xt1 i Xt2 i Xw1 i Xw2 i Xw3 i Xw4 i T The submatrices for the displacement vector are Xci ¼ ½ xci

yci

zci

uci wci ; Xtj i ¼ ½ xtj i ytj i ztj i htj i utj i wtj i 

hci

Xwk i ¼ ½ xwk i ywk i zwk i hwk i wwk i ; Mci ¼ diag½ mc mc mc Jcx J cy J cz 

5. Conclusions

Mtj i ¼ diag½ mt mt mt Jtx J ty Jtz ; Mwk i ¼ diag½ mw mw mw J wx J wz 

This paper establishes a refined 3D random wheel/rail contact model for a train-track-bridge system based on the random theory of PDEM and executes this model in the MATLAB@ platform. Considering random track irregularities and random wheel/rail contact geometry, the system dynamic responses and wheel/rail interaction forces can be analyzed using PDEM. The conclusions are as follows:

where j ¼ 1; 2; k ¼ 1; 2; 3; 4: In more detail, the submatrices in Eq. (8) can be expressed as

2

6 6 6 6 6 Kci ¼ 46 6 6 6 4

(a) The refined 3D random wheel/rail contact model clearly reveals the random dynamic characteristics of the wheel/rail interaction according to the time-varying probability density evolution functions of the wheel/rail contact locations. This theory can be effectively applied to guide track design, optimize wheel/rail tread design and improve the accuracy of wheel/rail dynamic calculations. (b) Compared with MCM, PDEM is 1–2 orders of magnitude more efficient for the random dynamic simulation of traintrack-bridge systems at the same accuracy. (c) The Chinese high-speed railway track spectrum is smoother than the German low-interference track spectrum, and the dynamic response amplitudes of the former are two to three times greater than those of the latter.

2 6 6 6 6 6 6 6 6 6 K t j i ¼ 26 6 6 6 6 6 6 6 6 4

2k1x þ k2x

k2x

k2z 2

2h3 k1y  h2 k2y

0

0

0 Þ

0 2

2

h2 k2x þ 2h3 k1x þ2L2t k1z syms

0

0

0

0

0

0

2

0

7 7 7 7 7 7 7 7 7 5

2

b3 k2x þ L2c k2y

0

!

3

7 7 7 7 0 7 7 7 7 0 7 7 7 7 7 7 0 7 7 !7 2 2 b3 k2x þ 2b4 k1x 7 5 þ2L2t k1y 0

2

2

0

3

3 0 0 0 h1 k2x 0 k2x 6 0 0 h2 k2y 0 0 7 k2y 6 7 6 7 6 0 7 0 0 0 0 k 2z 6 7 Kctj i ¼ 26 2 7 0 b1 k2z  h1 h2 k2y 0 0 7 h1 k2y 6 0 6 7 j 6 h k 0 ð1Þ Lc k2z 0 h1 h2 k2x 0 7 4 1 2x 5 2 0 ð1Þjþ1 Lc k2y 0 ð1Þ j h2 Lc k2y 0 b3 k2x

0

2

h1 k2y

2

2k1y þ k2y

þh2 k2y þ 2h3 k1y

0

syms

2h3 k1x  h2 k2x

ð

h1 k2x

2 h1 k2x þ L2c k2z

0

2

0

b1 k2z þ h1 k2y

0

b1 k2z þ 2b2 k1z

0

k2y 0

0

2k1z þ k2z

0

137

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

2

Kwk i

k1x

6 6 6 ¼ 26 6 6 4

0

0

0

0

k1y

0

0

0

k1z þ krail

0

0

2

b0 ðk1z þ krail Þ

Table 2 (continued)

7 7 7 7 7 7 5

0 2 b4 k1x

syms 2

Ktj wk i

3

k1x 6 0 6 6 6 0 6 ¼ 26 6 0 6 6 h k 4 3 1x 0

N5

0 k1y 0 h3 k1y

0 0 k1z 0

0 0 0 ð1Þkþ1 b0 b2 k1z

0 0 0 0

0

ð1Þk Lt k1z

0

0

ð1Þkþ1 Lt k1y

0

0

b4 k1x

0 0 0 0 0 N6

3 7 7 7 7 7 7 7 7 7 5

2

01ðNrL nik 12Þ f¼f

0 0 0 0 0

ik

UYik ¼ ½ 01nik 0 N 1 0 0 0 N 2 0 N 3 0 0 0 N 4 01ðNrL nik 12Þ f¼fik UikZ ¼ ½ 01nik 0 0 N1 0 N2 0 0 0 N 3 0 N 4 0 01ðNrL nik 12Þ f¼fik

Uwik ¼ ½ 01nik 0 0 0 0 0 N5 0 0 0 0 0 N6 01ðNrL nik 12Þ f¼fik 2

3

2

N1 ¼ 1  3ðf=lÞ þ 2ðf=lÞ ; N2 ¼ fð1  2ðf=lÞ þ ðf=lÞ Þ; 2

3

Units

Tractor

Trailer

mw (mass of the wheel-set) Jwx (roll mass moment of the wheel-set)

kg kg m2 kg m2 kN/ m kN/ m kN/ m kN/ m kN/ m kN/ m kN s/ m kN s/ m kN s/ m kN s/ m kN s/ m kN s/ m m m m m

2400 1200

2400 1200

1200

1200

9000

15,000

1040

700

3000

5000

240

280

400

300

480

560

50

0

50

50

30

30

60

120

60

60

30

25

24.775 17.375/2 1.25 1.00

24.775 17.375/2 1.25 1.00

m

0.95

0.95

m

1.00

1.00

m

0.95

0.95

m m

1.496/2 2.50

1.496/2 2.50

m

1.80

1.80

m

0.80

0.80

m

0.30

0.20

m m

-0.05 2.30

0.10 2.30

m

0.92/2

0.92/2

Jwz (yaw mass moment of the wheel-set)

where k1x , k1y and k1z are the primary suspension stiffnesses; k2x , k2y and k2z are the secondary suspension stiffnesses; Lc denotes the horizontal distance between the centers of gravity of the car body and each bogie; Lt denotes half of the bogie axle base; h1 is the height of the car body above the secondary suspension system, h2 is the secondary suspension system above the bogie, h3 is the bogie above the wheel-pair and h4 is the wheel-pair above the bridge centroid; and b1 and b2 denote the half-distances of the vertical spring-damper of the secondary and primary suspension systems, respectively, b3 and b4 denote the half-distances of the horizontal spring-damper of the secondary and primary suspension system, respectively, and b0 denotes the half-distance of the rail gauge. (2) Shape functions of the rail element The specific expression of the shape functions Um ik is UikX ¼ ½ 01nik

Vehicle parameters

2

N3 ¼ 3ðf=lÞ  2ðf=lÞ ; N4 ¼ fððf=lÞ  f=lÞ; N5 ¼ 1  f=l; N6 ¼ f=l; where fik (k = 1, 2, 3, 4) denotes the distance between the kth wheel-set of the ith vehicle and the left node of the rail element on which the wheel-set is acting.

k1x (longitudinal stiffness of the 1st suspension system, per side) k1y (lateral stiffness of the 1st suspension system, per side) k1z (vertical stiffness of the 1st suspension system, per side) k2x (longitudinal stiffness of the 2nd suspension system, per side) k2y (lateral stiffness of the 2nd suspension system, per side) k2z (vertical stiffness of the 2nd suspension system, per side) c1x (longitudinal damping of the 1st suspension system, per side) c1y (lateral damping of the 1st suspension system, per side) c1z (vertical damping of the 1st suspension system, per side) c2x (longitudinal damping of the 2nd suspension system, per side) c2y (lateral damping of the 2nd suspension system, per side) c2z (vertical damping of the 2nd suspension system, per side) L (full length of the vehicle) Lc (half-distance of the two bogies) Lt (half-distance of two wheel-sets) b1 (half-span of the 2nd vertical suspension system) b2 (half-span of the 1st vertical suspension system) b3 (half-span of the 2nd horizontal suspension system) b4 (half-span of the 1st horizontal suspension system) b0 (half-span of the wheel-set) e (lateral distance from the wheel-set to the bridge center) h (vertical distance from the rail to the bridge center) h1 (height of the body above the 2nd suspension system) h2 (height of the 2nd suspension system above the bogie) h3 (height of the bogie above the wheel-set) h4 (height of the wheel-set above the bridge centroid) Rw0 (initial rolling radius of the wheel)

Appendix B Table 2 Major parameters of the vehicle used in this study.

References

Vehicle parameters

Units

Tractor

Trailer

M c (mass of the body) Jcx (roll mass moment of the body)

kg kg m2 kg m2 kg m2 kg kg m2 kg m2 kg m2

48,000 115,000

44,000 100,000

2700000

2700,000

2700,000

2700,000

3200 3200

2400 2400

6800

6800

7200

7200

Jcy (pitch mass moment of the body) Jcz (yaw mass moment of the body) M t (mass of the bogie) Jtx (roll mass moment of the bogie) Jty (pitch mass moment of the bogie) Jtz (yaw mass moment of the bogie)

[1] Zhai WM, Cai CB. Train/Track/Bridge dynamic interactions: Simulation and Applications. Vehicle Syst Dyn 2002;37S:653–65. [2] Kalker JJ. Wheel-rail rolling contact theory. J Wear 1991;144:243–61. [3] Shen ZY, Hedrick JK, Elkins JA. A comparison of alternative creep force models for rail vehicle dynamic analysis. In: Hedrick JK, editor. Paper presented at the 8th IAVSD Symposium on Dynamics of Vehicles on Road and Tracks. Massachussetts, Cambridge: Swets and Zeithinger; 1983. p. 591–605. [4] Polach O. Afast wheel-rail forces calculation computer code. Vehicle Syst Dyn 1999:728–39. Suppl. 33. [5] Kalker JJ. A brief survey of wheel-rail contact Theory, Algorithms, Applications. Progress Ind Math ECMI 1997;96:60–76. [6] Shabana AA, Berzeri M, Sany JR. Numerical procedure for the simulation of wheel/rail contact dynamics. J Dyn Syst-T Asme 2001;123(2):168–78. [7] Kalker JJ. On the rolling contact of two elastic bodies in the presence of dry friction. The Netherlands: Delft University; 1967. Ph. D. Thesis. [8] Kalker JJ. A fast algorithm for the simplified theory of rolling contact. Vehicle Syst Dyn 1982;11:1–13.

138

Z.-w. Yu, J.-f. Mao / Engineering Structures 144 (2017) 120–138

[9] Pombo J, Ambrosio J, Silva M. A new wheel-rail contact model for railway dynamics. Vehicle Syst Dyn 2007;45(2):165–89. [10] Chen G, Zhai WM. A new wheel/rail spatially dynamic coupling model and its verification. Vehicle Syst Dyn 2004;41(4):301–22. [11] Shabana AA, Zaazaa KE, Escalona JL, et al. Development of elastic force model for wheel/rail contact problems. J Sound Vib 2004;269(1–2):295–325. [12] Song MK, Noh HC, Choi CK. A new three-dimensional finite element analysis model of high-speed train-bridge interactions. Eng Struct 2003;25 (13):1611–26. [13] Xia H, Zhang N. Dynamic analysis of railway bridge under high-speed trains. Comput Struct 2005;83(23–24):1891–901. [14] Li H, Xia H, Soliman M, et al. Bridge stress calculation based on the dynamic response of coupled train-bridge system. Eng Struct 2015;99:334–45. [15] Dimitrakopoulos EG, Zeng Q. A three-dimensional dynamic analysis scheme for the interaction between trains and curved railway bridges. Comput Struct 2015;149:43–60. [16] Antolin P, Zhang N, Goicolea JM, et al. Consideration of nonlinear wheel-rail contact forces for dynamic vehicle-bridge interaction in high-speed railways. J Sound Vib 2013;332(5):1231–51. [17] Kwark JW, Choi ES, Kim YJ, et al. Dynamic behavior of two-span continuous concrete bridges under moving high-speed train. Comput Struct 2004;82(4– 5):463–74. [18] Majka M, Hartnett M. Effects of speed, load and damping on the dynamic response of railway bridges and vehicles. Comput Struct 2008;86(6):556–72. [19] Yang YB, Yau DJ, Wu SY. Vehicle-Bridge interaction dynamics:with applications to High-Speed rail. Singapore: World Scientific Pub Co Inc; 2004. [20] Zhang N, Xia H, Guo W. Vehicle-bridge interaction analysis under high-speed trains. J Sound Vib 2008;309(3–5):407–25. [21] Guo WW, Xia H, De Roeck G, et al. Integral model for train-track-bridge interaction on the Sesia viaduct: Dynamic simulation and critical assessment. Comput Struct 2012;112:205–16. [22] Au FTK, Wang JJ, Cheung YK. Impact study of cable-stayed railway bridge with random rail irregularities. Eng Struct 2002;24(5):529–41. [23] Xia H, Zhang N. Dynamic Interaction of Vehicles and Structure (2nd edition), Bei jing. China: Science press; 2005. [24] Zeng QY, Guo XR. Train-bridge time-variant system analysis theory and application. Beijing: China Railway Press; 1999. [25] Kardas-Cinal E. Spectral distribution of derailment coefficient in Non-Linear model of railway Vehicle-Track system with random track irregularities. J Comput Nonlin Dyn 2013;8(0310143):1–9. [26] Rocha JM, Henriques AA, Calçada R. Probabilistic safety assessment of a short span high-speed railway bridge. Eng Struct 2014;71:99–111. [27] Salcher P, Pradlwarter H, Adam C. Reliability assessment of railway bridges subjected to high-speed trains considering the effects of seasonal temperature changes. Eng Struct 2016;126:712–24. [28] Mckay MD, Beckman RJ, Conover WJ. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 2000;21(1):239–45. [29] Naess A, Leira BJ, Batsevych O. System reliability analysis by enhanced Monte Carlo simulation. Struct Saf 2009;31(5):349–55. [30] Rocha JM, Henriques AA, Calçada R, et al. Efficient methodology for the probabilistic safety assessment of high-speed railway bridges. Eng Struct 2015;101:138–49. [31] Zhang ZC, Lin JH, Zhang YH, et al. Non-stationary random vibration analysis for train bridge systems subjected to horizontal earthquakes. Eng Struct 2010;32 (11):3571–82.

[32] Zhang ZC, Lin JH, Zhang YH, et al. Non-stationary random vibration analysis of three-dimensional train-bridge systems. Vehicle Syst Dyn 2010;48(4):457–80. [33] Zhang Y, Lin J, Zhao Y, et al. Symplectic random vibration analysis of a vehicle moving on an infinitely long periodic track. J Sound Vib 2010;329 (21):4440–54. [34] Zeng ZP, Zhao YG, Xu WT, et al. Random vibration analysis of train-bridge under track irregularities and traveling seismic waves using train-slab trackbridge interaction model. J Sound Vib 2015;342:22–43. [35] Muscolino G, Ricciardi G, Impollonia N. Improved dynamic analysis of structures with mechanical uncertainties under deterministic input. Probabilist Eng Mech 2000;15(2):199–212. [36] Li J, Chen JB, Sun WL, et al. Advances of the probability density evolution method for nonlinear stochastic systems. Probabilist Eng Mech 2012;28 (SI):132–42. [37] Li J, Chen JB. Stochastic dynamic of structures. Singapore: John Wiley & Sons (Asia) Pte Ltd; 2009. [38] Li J, Chen JB. The principle of preservation of probability and the generalized density evolution equation. Struct Saf 2008;30(1):65–77. [39] Yu ZW, Mao JF, Guo FQ, et al. Non-stationary random vibration analysis of a 3D train–bridge system using the probability density evolution method. J Sound Vib 2016;366:173–89. [40] Mao JF, Yu ZW, Xiao YJ, et al. Random dynamic analysis of a train-bridge coupled system involving random system parameters based on probability density evolution method. Probabilist Eng Mech 2016;46:48–61. [41] Chen JB, Sun WL, Li J, et al. Stochastic harmonic function representation of stochastic processes. J Appl Mech-T Asme 2013;80(0110011):1–11. [42] Chen J, Li J. Optimal determination of frequencies in the spectral representation of stochastic processes. Comput Mech 2013;51(5):791–806. [43] Zeng QY. The principle of total potential energy with stationary value in elastic system dynamics. J Huazhong Univ Sci Technol 2000;28(1):1–3. [44] Johnson KL. Contact mechanics. UK: Cambridge University Press; 1985. [45] Kalker JJ. Three-dmensional elastic bodies in rolling contact. The Netherlands: Kluwer Academic Publishers; 1990. [46] Zhai WM. Vehicle-Track coupling Dynamics(Third edition). Bei Jing: Science press; 2007. [47] Zhai WM, Xia H, Cai CB, et al. High-speed train-track-bridge dynamic interactions-Part I: Theoretical model and numerical simulation. Int J Rail Transp 2013;1:1–24. [48] Shinozuka M. Simulation of multivariate and multidimensional random processes. J Acoust Soc Am 1971;49(1):357–68. [49] Ontes RK, Enochson L. Digital time series analysis. New York: Wiley; 1972. [50] Salcher P, Adam C. Modeling of dynamic train–bridge interaction in highspeed railways. Acta Mech 2015;226(8):2473–95. [51] Fang K, Lin DKJ, Uniform design in computer and physical experiments. In: Tsubaki H, Nishina K, Yamada S, Tsubaki H, Nishina K, Yamada S., Editors. 2008: New York. p. 105–125. [52] Zhou Y, Fang K. An efficient method for constructing uniform designs with large size. Computation Stat 2013;28(3):1319–31. [53] The technical task book on the Inter City Express (ICE). 1993, Munich Research Center of German Federal Railway. [54] Hu ST, Niu B, Ke ZT, et al. Study on the optimization of standard span length simply supported box girder for high-speed railway. China Railway Sci 2013;34(1):15–21.